Receiver Cancellation Technique for Nonlinear Power...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007 2499

Receiver Cancellation Technique for NonlinearPower Amplifier Distortion in

SDMA–OFDM SystemsFernando Gregorio, Student Member, IEEE, Stefan Werner, Member, IEEE,

Timo I. Laakso, Senior Member, IEEE, and Juan Cousseau, Senior Member, IEEE

Abstract—Space-division multiple access (SDMA) and orthogo-nal frequency-division multiplexing (OFDM) can be combined todesign a robust communications system with increased spectralefficiency and system capacity. This combination is one of themost promising candidates for future wireless local area networkimplementations. However, one drawback of OFDM systems is thehigh peak-to-average power ratio, which imposes strong require-ments on the linearity of power amplifiers (PAs). Such linearityrequirements translate into high back-off that results in low powerefficiency. In order to improve power efficiency, a PA nonlinearitycancellation (PANC) technique is introduced in this paper. Thistechnique reduces the nonlinear distortion effects on the receivedsignal. The performance of the new technique is evaluated withsimulations, which show significant power efficiency improve-ments. To obtain meaningful results for comparison purposes, wederive a theoretical upper bound on the bit error rate perfor-mance of an SDMA-OFDM system subject to PA nonlinearities.In addition, a novel channel estimation technique that combinesfrequency- and time-domain channel estimation with PANC isalso presented. Simulation results show the robustness of thecancellation method also when channel estimation is included.

Index Terms—Multiuser, orthogonal frequency-division multi-plexing (OFDM), power amplifier (PA) nonlinearities cancellation,space-division multiple access (SDMA).

I. INTRODUCTION

S PACE-DIVISION multiple access (SDMA)–orthogonalfrequency-division multiplexing (OFDM) can be applied

in wireless local area network (WLAN) [1] systems to increasethe data rate and the system capacity. Accurate estimation ofthe uplink user channels enables multiuser detection techniquesfor user separation. Systems that have low or medium channelmobility, which is the case for WLAN systems, are well suited

Manuscript received March 27, 2006; revised August 19, 2006 andDecember 9, 2006. This work was supported in part by ALβAN, EuropeanUnion Programme of High Level Scholarships for Latin America, underContract E03D19254AR and in part by the Nokia Foundation. The review ofthis paper was coordinated by Prof. L. Lampe.

F. Gregorio, S. Werner, and T. I. Laakso are with the Signal Process-ing Laboratory, Smart and Novel Radios (SMARAD) Centre of Excellence,Helsinki University of Technology, 02015 Espoo, Finland (e-mail: [email protected]; [email protected]; [email protected]).

J. Cousseau is with the Consejo Nacional de Investigaciones Científicasy Técnicas (CONICET)–Department of Electrical and Computer Engineer-ing, Universidad Nacional del Sur, Bahía Blanca 8000, Argentina (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.899965

for that situation since the channel estimation process does notrequire high complexity.

The combination of several signals with different phasesand frequencies that are typical for OFDM systems causes alarge peak-to-average power ratio (PAPR) [2]. This can resultin considerable distortion effects when the composite signal isamplified by a power amplifier (PA), which typically has non-linear (NL) characteristics. The high PAPR of OFDM signalsrequire linear PAs with a high dynamic range. However, linearamplifiers tend to have low power efficiency [3], which leadsto a reduced battery life, which is a critical resource in mobilesystems.

Several techniques that combat NL effects at the transmitterside in OFDM systems have been proposed in the literature,e.g., PAPR reduction via mapping or coding [4], [5], linearscaling [6], clustered OFDM for low PAPR implementa-tions [7], allocation methods that minimize intermodulationproducts [8], and beamforming designs that employ PAPRconstraints [9]. Techniques that are applicable at the transmitterside also include predistorters (PDs) [10] that enable power-efficient amplifiers with reduced out-of-band emission and lowwaveform distortion. However, the rather high computationalcomplexity that is associated with PDs may prohibit theiruse in small mobile transceiver structures where low powerconsumption is required.

Two important aspects for reducing NL PA effects are out-of-band and in-band distortion. The in-band distortion degradesthe own bit error rate (BER) performance, whereas the out-of-band distortion affects users that are located in the adjacentfrequency bands.

The effect of NL PA on the bit error probability in anSDMA–OFDM system is an important issue that must beconsidered in a realistic system design. This paper derives anupper bound on the BER for a least squares (LS) detector in anSDMA–OFDM system subject to PA nonlinearities. Our resultsare not restricted to a particular PA model. Furthermore, weconsider the realistic case of low clipping levels, i.e., severalclipping events during an OFDM symbol. The assumption oflow clipping levels enables us to model the NL distortion as anadditive Gaussian noise (see [11]). We also verify the validity ofthis assumption for our setup through simulations. The solutionobtained also provides an upper bound for the achievableBER of more advanced receiver structures, like the minimummean square error (MMSE) and NL detectors [12], where thederivation of BER expressions is far more complicated.

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2500 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

Fig. 1. Block diagram of an SDMA–OFDM system that consists of one base station equipped with P antennas and L mobile users, each with a singleantenna element.

The BER upper bound also motivates us to introduce a newcancellation technique that reduces the NL effects. To reducethe NL distortion, we propose the combination of multiuserdetection and NL distortion mitigation techniques at the re-ceiver side. The basic idea of the proposed PA nonlinearitycancellation (PANC) technique is given as follows. With aninitial estimate of the user symbols, the distortion effects canbe estimated if the PA model is known. After that, the NLdistortion can be removed from the received signal, and newand improved symbol estimates can be obtained. This proce-dure can be repeated in an iterative manner to obtain almostundistorted estimates in two or three iterations. This conceptualidea was used in [13] in a single-user wireline system usingadaptive OFDM with a large number of carriers. In practice,PA model parameters need to be estimated and sent to thereceiver. Our simulations confirm the good performance of thePANC technique for the cases of known and estimated PAmodel parameters. In our simulation studies, the nonlinearity ismodeled as a simple static nonlinearity. However, we stress thatthe PANC technique can be used with more general models thatinclude memory effects, e.g., Wiener models [10], [14], [15].

Another important issue, which is by far not addressed inthe literature, is channel estimation in systems that are affectedby nonlinearities. In addition to PANC, we propose a multiuserchannel estimation technique that gives accurate results whenPA nonlinearities are considered. The novel channel estima-tion method is combined with PANC for quasi-stationary andtime-varying channels. We show that the combination givesmean square error (MSE) levels for the channel estimatesthat approach the values obtained with linear amplifiers. Theevaluation of PANC for low mobility systems and systemswith moderate mobility is included to illustrate the proposedmethods in a practical context.

In summary, the contributions of the paper are listed asfollows:

1) a theoretical BER upper bound for SDMA–OFDM sys-tems with PA nonlinearities;

2) extension of the PANC technique in [13] to the case ofmobile SDMA–OFDM systems;

3) a new frequency- and time-domain (FD–TD) channelestimation scheme compatible with the PANC technique.

The paper is organized as follows. Section II reviews con-cepts of multiuser SDMA–OFDM systems and multiuser de-tection techniques and introduces the notation to be used in

the remaining parts of the article. The theoretical BER boundfor SDMA–OFDM systems with PA nonlinearities is derivedin Section III. The new PANC technique for the multiusercase is introduced in Section IV. Section V presents a newFD–TD channel estimation approach tailored for the proposedPANC technique. Section VI considers some practical issuesthat are related to the implementation of the proposed method.Section VII provides simulations of system efficiency, out-of-band distortion analysis, MSE channel estimation, and BER.Finally, conclusions are drawn in Section VIII.Notation: In this paper, small boldface letters are used to

denote vectors, and capital boldface letters are used to denotematrices. In addition, the standard font (Times Roman) is usedfor time-domain (TD) variables, and calligraphic letters areused to denote frequency-domain (FD) variables. For example,H, h, and h denote a matrix, a vector, and a scalar variable inthe TD, respectively. Their corresponding notations in the FDare H , h, and h, respectively.

II. MULTIUSER MULTIPLE-INPUT–MULTIPLE-OUTPUT

(MIMO)–OFDM SYSTEMS

This section presents the SDMA–OFDM system model thattakes into account PA effects. Thereafter, we briefly reviewthe most common receiver techniques for separating the usersignals at the base station. Of particular interest is the LSdetector, which is used in the BER and capacity studies inSections III and IV, respectively.

A. System Model

The multiuser SDMA–OFDM system under considerationhas N subcarriers and consists of one base station that isequipped with P antennas and L mobile users with a singletransmit antenna. That results in a P × L MIMO–OFDM sys-tem. It is assumed that all users are simultaneously transmittingindependent signals on all N subcarriers.

A block diagram of the system is shown in Fig. 1, wherethe cyclic prefix insertion/elimination blocks for combattingintersymbol interference are denoted CP and RCP, respectively.

The transmitted signal from user j at time instant n isgiven by

xj(n) = Gcpxj(n) = GcpQNxj(n) (1)

GREGORIO et al.: RECEIVER CANCELLATION TECHNIQUE FOR PA DISTORTION IN SDMA–OFDM SYSTEMS 2501

where Gcp is the (N + v)×N cyclic prefix insertion matrix[16], v is the length of the cyclic prefix,N + v is the total lengthof the OFDM symbol, x(n) is the inverse discrete Fouriertransform (IDFT) of the modulated symbols xj(n) ∈ C

N×1

without the cyclic prefix, and QN is the N ×N IDFT matrix.The multicarrier signal after passing the NL PA g[·] can be

written as

xgj (n) = g [xj(n)] = KLxj(n) + dj(n) (2)

where the first term xj(n) is the distortion-free discrete-timeinput signal vector of (1), and KL is the gain of the linearpart. The second term dj(n) is the NL distortion, which isa function of the modulated symbol vector xj(n) and the PAtransfer function g[·] (details of some common NL PA modelsare presented in Appendix A). The value of KL approachesunity for clipping levels higher than 6 dB [17], i.e., when theconstellation scaling is insignificant.

The received signal at antenna i after removing the cyclicprefix yi(n) is formed by the superposition of the indepen-dently faded signals that are associated with the L users sharingthe same space–frequency resource. The received signal, whichis assumed to be corrupted by circular complex Gaussian noiseat the array elements, is given by

yi(n) =L∑

j=1

[KLHi,j(n)xj(n) +Hi,j(n)dj(n)] + ni(n)

(3)

where Hi,j(n) is an N ×N circulant TD channel matrix attime instant n, which is formed by the channel response vectorhi,j(n) for the link between user j and base station antenna i.The FD expression of the received signal is obtained by takingthe discrete Fourier transform (DFT) of (3).

Let y(n, k) = [y1(n, k), . . . , yP (n, k)]T denote the vector ofreceived signals at each antenna on subcarrier k. Then, thereceived signal vector for each subcarrier can be written as

y(n, k) = KLH(n, k)x(n, k) +H(n, k)d(n, k) + n(n, k)(4)

where H(n, k) ∈ CP×L is the channel transfer matrix,

x(n, k) = [x1(n, k), . . . ,xL(n, k)]T is the vector thatcontains transmitted signals from each user, d(n, k) =[d1(n, k), . . . , dL(n, k)]T is the vector that contains the NLdistortion of each user on subcarrier k, and n ∈ C

P×1 is theadditive noise assumed to be circular complex Gaussian withE[nnH ] = σ2

nI.The FD channel transfer matrix H(n, k) in (4) is given

by [12]

H(n, k) =

h1,1(n, k) h1,2(n, k) · · · h1,L(n, k)h2,1(n, k) h2,2(n, k) · · · h2,L(n, k)

......

. . ....

hP,1(n, k) hP,2(n, k) · · · hP,L(n, k)

where hi,j(n, k) denotes the channel response on subcarrierk at time n between antenna element i of the base stationand user j.

B. FD Detectors

An estimate x(n, k) of the L user transmitted signals x(n, k)can be obtained by linearly combining the signals at the Preceive antennas using a weight matrix W ∈ C

P×L as follows:

x(n, k) = WHy(n, k). (5)

The standard LS combiner W = W LS is given by [18]

W LS =1KL

H(n, k)[HH(n, k)H(n, k)

]−1. (6)

Alternatively, an MMSE detector can be used. This alterna-tive exploits the available statistical knowledge of the noise andmakes a tradeoff between the multiuser interference (MUI) andthe measurement noise. The optimal MMSE weight is obtainedas follows:

W MMSE =1KL

[H(n, k)PρH

H(n, k) + σ2nI]−1

H(n, k)Pρ

(7)

where matrix Pρ is an (L× L) diagonal matrix of the formdiag(ρ1, . . . , ρL), with ρj = σ2

j /σ2n being the signal-to-noise

ratio (SNR) of user j.Other options for user separation include NL techniques, e.g.,

parallel interference cancellation (PIC) and successive interfer-ence cancellation (SIC) [12]. This paper will only consider theuse of the linear LS and MMSE receivers.

III. BER UPPER BOUND IN AN SDMA–OFDMSYSTEM WITH AN NL PA

In this section, a theoretical BER analysis is carried out forthe case when an LS receiver is used to separate the trans-mitted multiuser signal. The obtained results provide an upperbound for the performance of different receiver structures. Inparticular, the LS performance will approach that of the MMSEdetector for high SNRs.

In the analysis, we use the following assumptions.

A1) We have low and medium clipping levels, i.e., theinput saturation voltage of the PA is fixed at a lowlevel, creating several clipping events during an OFDMsymbol.

A2) The FD channel coefficients hi,j(n, k) are assumedto be independent stationary zero-mean unit variancecircular complex Gaussian-distributed processes.

A3) The LS detector defined in (6) is used for userseparation.

A4) A low channel noise approximation is used.

Assumption A1) enables us to consider the distortion term in(3) to be additive Gaussian noise with a variance that is equalto σ2

d (see [11] for details).To validate assumption A1), Fig. 2 shows a Gaussian com-

plementary cumulative distribution function (CCDF) with anadequate variance, and the simulation results for an OFDMsystem with N = 512 subcarriers and QPSK modulation. Ascan be observed, the curves show good agreement when


Fig. 2. CCDF of distortion noise for the SSPA model with p = 2.

clipping levels are lower than 6 dB. The clipping level ν isdefined as

ν =As√

E{|x(n)|2

} (8)

where As is the amplifier input saturation,√E{|x(n)|2} is

the root-mean-square value of the OFDM signal. From theabove, we conclude that the distortion noise can be consideredas additive Gaussian noise if clipping levels are lower than6 dB. Higher clipping levels can be analyzed using the impul-sive noise model proposed in [19].

Based on assumptions A1)–A4), the derivation of a BER up-per bound follows an approach that is similar to that presentedin [20], in which the authors present an analysis of the errorprobability for a system with antenna diversity.

Applying the LS detector to the signal model of (4), thefollowing estimate of the transmitted signal is obtained:

xLS(n, k) =WHLS [H(n, k)x(n, k)

+H(n, k)d(n, k) + n(n, k)]

=x(n, k) +d(n, k)KL

+WHLSn(n, k). (9)

The LS estimate xLS(n, k) is a noisy estimate of theoriginal vector x(n, k) and can be modeled by a Gaussiandistribution with mean value µ and covariance matrixRLS, where µ= Ex{xLS(n, k)}= x(n, k), and RLS=Ex{[WH

LSn(n, k)] + d(n, k)/KL] [WHLSn (n, k) + d(n, k)/

KL]H} = σ2n[H

H(n, k)H(n, k)]−1/K2L + σ2

d/K2L (Ex de-

notes conditional expectation with respect to x).In the following equations, time index n and frequency index

k are dropped to simplify the notation. Under assumption A2),the MSE of the LS receiver for user j is given by [12]

ξj =

[HH(n, k)H(n, k)

]−1

j,jσ2n + σ2

d

K2L

(10)

where 1/[HHH]−1j,j , which corresponds to the inverse of ele-

ment (j, j) of matrix [HHH]−1, is a scalar random variablewith a chi-square distribution with 2(P − L+ 1) degrees offreedom [20]. Defining γ = (1/[HHH]−1

j,j)/σ2n as the instan-

taneous SNR, the MSE can be written in a compact form as

ξj =σ2a

K2Lγ+

σ2d

K2L

(11)

where σ2a is the energy of the transmitted complex-valued data

symbols.The conditional error probability Pe(E|γ) and the MSE can

be related by [20]

Pe(E|γ) ≤ exp(− 1ξj

)(12)

where the approximation is valid for low noise levels[see Assumption A4)]. Combining (11) and (12) results in thefollowing upper bound on the conditional error probability:

Pe(E|γ) ≤ exp(− K2

Lγ

σ2dγ + σ2

a

). (13)

To determine the average bit error probability, the conditionalprobability of error must be averaged over the fading channelstatistics as

Pe ≤∞∫

0

Pe(E|γ)P (γ)dγ (14)

where P (γ) is the statistics of the channel, which is a chi-squareprobability density function (pdf) with 2(P − L+ 1) degreesof freedom given by

P (γ) =1

(P − L)!γP−L+1γP−L exp

(−γ

γ

)(15)

and γ = E[γ] is the average SNR. Substituting the pdf (15)and the conditional bit error probability (13) in the integral(14) gives

Pe ≤1

(P − L)!γP−L+1

×∞∫

0

γP−L exp(− K2

Lγ

σ2dγ + σ2

a

− γ

γ

)dγ. (16)

Using a Taylor expansion of the exponential function in (16),the error probability can be expressed as

Pe ≤1

(P − L)!γP−L+1

∞∑m=0

(−1)m(m+ P − L)!m!

×(1σ2d

)m+P−L+1

K2mL UP−L+m+1,P−L+2

2

(σ2a

σ2dγ

)(17)


where UP−L+m+1,(P−L+2/2)(σ2a/σ

2dγ) is the Confluent Hyper-

geometric Function [21] defined as

Ua,b(z) =1Γ(a)

∞∫0

exp(−zt)ta−1(1 + t)b−a−1dt. (18)

Extensive simulations show that a close fit to the infinitesum of the original expression (see Appendix B) is obtainedby employing only the first 20–30 terms.

IV. PANC TECHNIQUE

In this section, we propose a PANC technique to combat thein-band distortion that is created by an NL PA.

We start by studying the effect of PA-induced NL distortionon the system capacity. A closed-form capacity expressionis obtained for the LS receiver. After that, the novel PANCtechnique is developed for use in an LS receiver structure.

A. NL PA Effects on System Capacity

Let us drop the time and frequency indices in (4), and let wdenote the additive distortion, i.e., w =Hd+ n. The interfer-ence vector w is uncorrelated with the transmitted signal. Thechannel capacity for the system in (4) can be written as [22]

C = EH

[log2

{det(Rw +HRxH

H)det(Rw)

}](19)

where the expectation operation EH [·] is over the randomchannel matrix H , Rx is the correlation matrix of the trans-mitted signal vector given by Rx = K2

Lσ2j IL. The correlation

matrix Rw of the uncorrelated interference terms, namely,channel noise and NL distortion, can be expressed as Rw =EH [(Hd+ n)(Hd+ n)H ] = σ2

dHHH + σ2nIP . Therefore,

substitution in (19) and using singular value decomposition forthe product HHH [23], the capacity becomes

C =E

log2

rank(HHH)∏i=1

(1 +

K2Lσ

2jλi

σ2n + σ2

dλi

)

=E

rank(HHH)∑

i=1

log2

{1 +

K2Lσ

2jλi

σ2n + σ2

dλi

} (20)

where λi is the ith eigenvalue of HHH , E[·] is the expectationover the eigenvalues λi, and (·) denotes the rank of the matrix.

From (20), we see how the capacity is reduced when the NLdistortion is considered. This is because λi is positive, and thelogarithmic function is monotonically increasing. Note also thatwhen σ2

d increases, KL decreases.In case a least squares receiver is employed, the capacity will

be affected by noise enhancement. Following the same steps as

above, the capacity for user j, assuming an LS equalizer, can bederived from (9) as follows:

CLS =E

[log2

{1 +

σ2j

σ2n[H

HH]−1j,j + σ2

d/K2L

}]

=E

[log2

{1 +

σ2jγ

σ2a + σ2

dγ/K2L

}](21)

where the instantaneous SNR γ = (1/[HHH]−1j,j)/σ

2n is

the same chi-square-distributed variable with 2(P − L+ 1)degrees of freedom that was used in the BER derivation ofSection III. The capacity is obtained by integrating (21) overthe chi-square distribution defined in (15). This integral canbe numerically evaluated. However, assuming a high SNRapproximation, the capacity can be solved for in closed formas (see Appendix C)

CLS ≈ log2

(1 +

1σ2d′

)− σ2

a

ln(2)

×[ln(β) + γe − exp(β)

([1 + βu(P − L)]

× Ei(−β) +P−L∑k=2

βkΓI(1− k, β)k

)](22)

where β = (σ2a/(σ

2d/K

2L))γ, γe is the Euler–Mascheroni con-

stant (γe = 0.5772 . . .) [24], Ei(·) is the exponential integralfunction, and ΓI(·) is the incomplete gamma function [21].The function u(P − L) is the unit step function defined asu(x) = 1 ∀x > 0 and u(x) = 0 ∀x ≤ 0.

Equation (22) was evaluated for L = 1 and L = 2 users andP = 4 for clipping levels of ν = 1 dB, ν = 2 dB, and ν = 4 dB,and the results are illustrated in Fig. 3. The exact capacityvalues were obtained using numerical evaluation. We see that(22) provides a tight approximation for SNR levels higher than5 dB. In order to evaluate the effect of NL distortion, the linearcase was also included. The asymptotic capacity is also plottedin Fig. 3 and is given by

C∞LS = lim

γ→∞CLS = log2

(1 +

K2L

σ2d

). (23)

It is clear from the expression for C∞LS that the capacity will

be bounded due to the NL distortion.

B. PANC Technique

The previous basic analysis shows that system capacity isstrongly affected by NL distortion. The result motivates theintroduction of a new cancellation technique in order to keepsystem capacity close to that of the linear case.

Assuming that the transmitter nonlinearity is known at thereceiver, the receiver can compute and estimate d(n) from thereceived vector x(n, k). An initial estimate of vector x(n, k)can be used to calculate the distortion vector d(n, k). Theestimated distortion vector is removed from the original re-ceived vector, and a new estimation of x(n, k) can be obtained.


Fig. 3. Capacity evaluation for an LS receiver with L = 1 and L = 2 users, P = 4 antennas, and clipping levels of ν = 1 dB, ν = 2 dB, and ν = 4 dB.The exact value of capacity is obtained using numerical integration, the high SNR approximation is computed using (22), and the linear case was also includedfor comparison.

This second estimate x(n, k) can be used to re-estimate thedistortion vector. This process can be iteratively performed untilsome specified bound is reached.

Using the LS detector of (6), the estimation of the jth usersignal is given by

xj(n, k) =1KL

[H(n, k)colj

×{[

H(n, k)HH(n, k)]−1

}]Hy(n, k) (24)

where colj{·} denotes the jth column vector of the correspond-ing matrix. Using xj , an estimate zj of the original transmittedconstellation xj is obtained by applying hard decoding (or moreefficient methods if coding is employed). This process is carriedout for all active carriers. Using the recovered symbols, the TDsignal is reproduced via IDFT as xj(n) = QNzj(n).

Assuming that the NL model of the PA, i.e., g[·], is knownat the receiver (details of g[·] are discussed in Section VI andAppendix A), the distortion term can be obtained using (2) asfollows:

dj(n) = g[x(m)j (n)

]−KLx(m)

j (n) (25)

where x(m)j (n) is the TD representation of the recovered signal

for user j at iteration m. The FD distortion term is obtainedapplying the DFT operator

dj(n) = QHN

{g[x(m)j (n)

]−KLx(m)

j (n)}. (26)

The distortion dj(n) = [dj(n, 1) · · · dj(n,N)]T is sub-

tracted from the estimated signal x(m)j . Using this result, the

transmitted constellation is re-estimated in a new decoding/distortion cancellation step. The process can be carried out iter-atively. Our simulation study, which is presented in Section VI,suggests that two or three iterations are usually sufficient.

The PANC technique for an SDMA–OFDM system with anLS receiver is summarized in Table I. Alternatively, the PANCtechnique can be combined with an MMSE receiver, whichestimates user signal j using (7).

V. PANC WITH CHANNEL ESTIMATION

The PANC technique proposed in Section IV assumes perfectknowledge of the channel. In practice, the channel needs tobe estimated, and estimation errors will degrade the overallperformance. There are two fundamental approaches for chan-nel estimation. One approach assumes that a whole OFDMsymbol is periodically transmitted, i.e., pilots are transmittedon all subcarriers. The second approach uses a reduced set ofdedicated subcarriers for pilot data, and the channel estimationis performed using interpolation. The second approach, whereeach user has its own dedicated pilot subcarriers, is addressedin this paper.

Channel estimation for the stationary case is executed in theinitialization. The obtained estimates can be used for relativelylong time until the channel conditions are modified. In thecase of time-varying channels, tracking capability must beconsidered, and the channel estimation process needs to becarried out for each OFDM symbol.

In the following, we discuss issues that are related to thechoice of the training symbols and the number of pilot subcarri-ers employed for systems with NL PA. The objective is to definean adequate number of pilot subcarriers, which reduces the NLPA effects on channel estimates. Finally, a channel estimationtechnique with tracking capabilities to combat NL distortionand MUI is introduced.

The novel combination of FD and TD channel estimation inaddition to PANC is developed in Section V-B. The main moti-vation behind a combined TD and FD scheme is to improve theestimation accuracy for a normal overhead (i.e., pilot carriers).If only FD channel estimation was used, the performance ofthe system will be severally degraded when a normal overhead


TABLE INL CANCELLATION TECHNIQUE (PANC) ALGORITHM

is used. In order to improve the accuracy, more pilot tones arerequired, which reduces the effective data rate of the system.The proposed combination of FD and TD channel estimationtechniques that are presented in Section V-B outperforms theresults that are obtained with conventional FD techniques, asshown in [25] and [26].

A. Channel Estimation Considerations

It is known that PA nonlinearities have a strong impact on thechannel estimation process [27]. Therefore, it is important thattraining symbols are not affected by nonlinearities in order toobtain good channel estimates. One option, which is applied inHIPERLAN II, is to design training symbols with low PAPR.

In the case of an SDMA–OFDM system, a group of equallyspaced subcarriers can be assigned to each user. With Nnonzero subcarriers and L users, a maximum of N/L subcar-riers can be allocated to each user, i.e., all users transmit theirtraining symbols on nonoverlapping subcarriers. The advantageof this method is that each user has only a group of N/L activesubcarriers during the training period, which reduces the PAPRof the OFDM signal [7]. This is explained by the fact that the

pdf of the OFDM signal with a large number of carriers can bewell approximated with a Gaussian distribution N (Nµ,Nσ2).Therefore, if only N/L carriers are active, the variance ofthe OFDM signal is reduced, and the PA mostly operates inthe linear region. Thus, the effect of nonlinearities is reducedin the channel estimation process.

However, there is a tradeoff between the number of pilotcarriers (maximum N/L), the channel estimation accuracy,and the level of NL distortion. The accuracy of the channelestimation is related to the number of subcarriers that areused for training, i.e., the interpolation error is reduced if thenumber of subcarriers is increased. Therefore, from a channelestimation point of view, the more pilot carriers used, the better.On the other hand, increasing the number of subcarriers leadsto an increased PAPR, i.e., increased NL distortion. Therefore,a total of T × L < N subcarriers are reserved for pilot data,where T is chosen to trade off the estimation accuracy anddistortion level.

Fig. 4 illustrates the pilot carrier allocation strategy for asystem withL = 4 users. In the figure, both stationary and time-varying channels are considered. For the case of a stationarychannel, the OFDM symbol includes only pilot carriers (shown


Fig. 4. Pilot carrier allocation. P: Pilot subcarriers. d: Data subcarriers.(a) Stationary channel. (b) Time-varying channel.

with P in the figure). In this case, as previously commented, thechannel estimation process is performed during the initializa-tion. For time-varying channels, where the channel estimation isperformed for each OFDM symbol, these symbols include pilotsubcarriers and data subcarriers (shown with d in the figure).The level of overhead is then equal to O = (TL/N).

Our simulation results suggest that T = 32 pilot subcarriersper active user is a good value for a system with L = 4 usersand N = 512 subcarriers, considering low clipping levels instationary channels. In the case of time-varying channels, wheretracking capabilities need to be considered, the number of pilotsubcarriers is reduced to T = 16 in order to obtain a reducedlevel of overhead.

B. FD–TD Channel Estimation With PANC

In this section, an iterative channel estimation technique thatcombines FD–TD channel estimation with PANC is discussed.The technique is applicable to estimation of both stationary(e.g., WLAN) and nonstationary channels (e.g., mobile sys-tems). The basic idea is to use the equalized signals from theFD processing to remove the NL distortion (and MUI in caseof nonstationary channels) and improve the channel estimatein TD.

The combined FD–TD channel estimation with PANC can besummarized in the following two steps:

Step 1) FD channel estimation with PANC. For the FDchannel estimation, we will assume that each useris given a set of T dedicated pilot carriers ac-cording to Fig. 4. For stationary channels, pilotcarriers are used only during the initialization. LetTj(kj,1, . . . , kj,T ) denote the set that specifies theT pilot carriers of user j. The channel frequencyresponse coefficients can be estimated over thesesubcarriers as in a single-user system. In the case ofa stationary channel, only pilot data are transmitted(see Fig. 4). This allows us to exactly reproduce thetransmitted TD signal and estimate the NL distortionusing (21). Thereafter, channel estimation may becarried out on the pilot subcarriers (k ∈ Tj) usingthe following expression:

hi,j(n, k) =yi(n, k)

KLxj(n, k) + dj(n, k)(27)

where yi(n, k) is the received signal at antenna i onsubcarrier k ∈ Tj , and xj(n, k) is the training sym-bol that is transmitted by user j at the correspondingsubcarrier.

Collecting the channel estimates obtained fromthe pilot carriers for user j in vector h

c

i,j(n) =[hi,j(n, kj,1), . . . , hi,j(n, kj,T )]T , the whole chan-nel FD response can be obtained through interpo-lation using truncated DFT matrices [28], i.e.,

hi,j(n) = QN

[QH

TjQTj

]−1

QHTj

hc

i,j(n) (28)

where QTjis formed by the T columns of the IDFT

matrix QN associated with the initially estimatedsubcarriers and the Lh rows associated with thenonzero TD channel taps.

In case of a nonstationary channel, pilot symbolsare tone multiplexed with useful data (see Fig. 4).This means that the user symbols on the data carriersneed to be estimated before we can construct theTD signal that is required for estimating dj(n, k)through (21). Therefore, an a priori channel estimatethat enables us to detect each user signal is required(here, we use the estimate hi,j(n− 1, k)). Afteracquiring an estimate of the NL distortion, (27) isused to obtain the new channel estimates over thepilots k ∈ Tj at time n. Finally, (28) provides thechannel for all subcarriers. Using the FD channelestimate, each user’s signal is detected and decodedto obtain zj(n) ∈ C

N , j = 1, . . . , L.Step 2) TD channel estimation with PANC. In this step, TD

channel estimation is performed. With NL distortionestimates {dj(n)}Lj=1 and detected user symbols{zj(n)}Lj=1 (or known pilots in case of stationarychannel) obtained from Step 1), we can now can-cel their effect from the TD received signal yi(n).Hence, we have

yi(n) = yi(n)− QN

×L∑

j=1;i�=j

Λi,j(n)[zj(n) + dj(n)] (29)

where Λi,j(n) = diag[hi,j(n)] is a diagonal matrixthat contains the FD channel estimates obtained inthe FD step.

After removal of the MUI and NL distortion, TDchannel estimation can now be performed on thesingle-user signal yi(n) using an LS approach [26],[29], i.e.,

hi,j(n) =[CH

j (n)Cj(n)]−1

CHj (n)yi(n) (30)

where Cj is the N × Lh circulant matrix formed bythe TD transmitted vector xj (i.e., each column ofCj equals the previous column rotated downward byone element [23]). A new iteration can be performedin TD in order to improve the channel estimationaccuracy [29].


Fig. 5. FD–TD channel estimation with PANC. s0: Transmitted symbols are recovered using the previous channel estimate. s1: Transmitted symbols arerecovered using the new FD channel estimate obtained after NL distortion cancellation (after PANC). s2: Transmitted symbols are recovered with a refined TDchannel estimate.

A block diagram of the FD–TD channel estimation withPANC is illustrated in Fig. 5.

Finally, some remarks about the application of the aforemen-tioned channel estimation technique in stationary and nonsta-tionary channels are given as follows. In a stationary channel,the estimation of the NL distortion is not effected by MUI (dueto known pilots). Furthermore, the level of NL distortion islower during the initial channel estimation process because onlyT out of N carriers are used, resulting in a reduced PAPR (seeSection V-A). For the case of nonstationary channels, the pilotcarriers are tone multiplexed with data carriers. This increasesthe NL distortion component in the channel estimation (dueto larger PAPR). Furthermore, an estimate of the MUI on thedata carriers is required in order to reconstruct the TD signalused for estimating the NL distortion. Therefore, as the a priorichannel estimate is based on the previous time instant, it isexpected that the decision errors will increase when the chan-nel variation increases (higher mobile speeds). Our simulationresults presented in Section VI confirm that the combination ofFD–TD channel estimation and PANC is feasible for stationaryand moderately time-varying channels. The FD–TD channelestimation with PANC is summarized in Table II.

VI. PRACTICAL CONSIDERATIONS

An attractive feature of the PANC technique is that it isimplemented at the base station receiver where more resourcesare available than in the mobile terminal. Furthermore, thehandset hardware does not need to be modified.

The PANC technique assumes the PA model of the user tobe known at the base station. The transmission of PA parame-ters must be included in the system initialization jointly withthe channel estimation and synchronization. As discussed inAppendix A, only one parameter (clipping level) must beknown for the limiter model. The clipping level and the smooth-

ness factor are required for the solid-state power amplifier(SSPA) model. These parameters can be estimated in theinitialization process before the channel estimation. A morerealistic alternative is to use a polynomial approach to model thenonlinearity, i.e.,

g[x(t)] =P∑j=0

aj |x(t)|j (31)

where x(t) is the input signal, g[x(t)] is the output signal ofthe PA model, and {aj}Pj=0 represents the polynomial coeffi-cients, with P being the polynomial order. By exciting the PAwith a power-swept single-tone signal, an LS estimate of thecoefficients aj can be obtained (see [30] for details). In orderto have a complete description of the PA at the base station,all coefficients of the polynomial model are required. We stressthat the PANC technique can be easily implemented with moregeneral PA models, including memory effects. However, this isbeyond the scope of this paper.

VII. SIMULATION RESULTS

In Sections VII-A–C, a complete performance study is pre-sented, where the channel estimation technique with PANC isevaluated in terms of MSE, BER, and efficiency for stationaryand nonstationary channel environments. The applicability ofPANC in WLAN systems is also tested. Finally, we comparethe BER results of PANC with a solution that uses a PDin the transmitter. In addition, we evaluate the performanceimprovement achieved with a solution that combines PANC atthe receiver and PD at the transmitter.

In our simulations, we assume P = 4 receive antennas andL = 1–4 users with a single transmit antenna that is equippedwith an SSPA with smoothness factor p = 2. The number ofsubcarriers employed is N = 512, and the length of the cyclic


TABLE IIFD–TD CHANNEL ESTIMATION WITH THE PANC ALGORITHM

prefix is v = 8. The modulation employed is either QPSK or16-quadratic-amplitude modulation (16-QAM). The channelis Rayleigh fading, with independent propagation paths, eachgenerated according to Jakes’ Doppler spectrum, with an expo-nential delay profile. The subcarrier frequency is 5 GHz, and abandwidth of 20 MHz is used. For the stationary channel, themobile speed is set to v(t) = 5 km/h, and v(t) = 30 km/h fornonstationary channels. The channel estimation for stationarychannels uses T = 32 subcarriers per user in the initializa-tion process, and T = 16 subcarriers per user in the case of

nonstationary channels. In all simulations, an LS detector isused for separating the user signals.

To characterize NL distortion and efficiency, the evaluationsthat are performed take into account the following parameters:

1) Input back-off (IBO) is defined as the ratio of the averagepower at the PA input and the input saturation power. Itcan be represented in decibels as IBOdB = 10 logA2

s −10 log ε2σ2

s , where σ2s , is the average power at the PA

input, and A2s is the input saturation power. Parameter


Fig. 6. BER versus Eb/N0 of a P × L SDMA–OFDM system with QPSK modulation for the case of (a) linear PA and (b) NL PA with clipping level ν = 1 dBand an LS detector. The mobile speed is set to v(t) = 5 km/h. The curves are plotted for P = 4 receive antennas, and L = 1, 2, 3, and 4. The upper bound of(17) is valid for all SNRs.

ε < 1 is used for scaling the symbols in order to reducethe clipping probability and limit the in-band distortion.The IBO employed determines the power that is drivenby the PA. Large values of IBO reduce the clipping prob-ability and decrease the BER degradation but also reducethe power efficiency (the dynamic range is decreased).In the simulations (Fig. 7), we use a class A PA that istypical for OFDM systems, whose power efficiency curveis given in [31] and [32].

2) Total degradationΨ is a measure for the balance betweenthe level of degradation and power efficiency, whichis defined as ΨdB = SNRNL

dB − SNRLdB + IBOdB, where

SNRNLdB = Eb/N0 is the SNR required to obtain a fixed

BER target in the presence of PA nonlinearities with afixed IBO, and SNRL

dB expresses the SNR required inthe case of linear PA. The BER target is typically setto 10−4.

A. Verification of the BER Analysis

Fig. 6 illustrates the BER results of the LS detector obtainedby theory and simulations. QPSK modulation is employed, andthe clipping level of the SSPA model is set to ν = 1 dB. TheBER curve for a linear PA is included for comparison purposesin Fig. 6(a).

As expected, the simulation results show a lower BER thanthe theoretical upper bound. For high SNR values, the BERbound is tighter than in the low-SNR region.

Note that if the upper bound derived in [20] were used(assuming linear PA), the predicted BER will be too optimisticand cannot be reached in practice. In that case, the systemdesign would be based on an erroneous limit. We believe thatthe upper bounds derived in Section III are more realistic for apractical system design.

B. Evaluation of PANC for Stationary Channels

The system considered in this section uses 16-QAM modula-tion on each carrier. We present the results obtained with PANC,assuming perfect channel state information (CSI) and using theFD–TD channel estimation approach proposed in Section V.

The BER versus SNR curves are shown in Fig. 7. As can beseen, a significant improvement is obtained with PANC. Therobustness of the PANC technique to channel estimation errorsis evident, showing BER results that are close to those obtainedwith perfect channel knowledge.

Total degradation curves are shown in Fig. 8 for L = 1 andL = 2 active users. From these curves, it is possible to concludethat the best operation points for a system with PANC and onewithout PANC are given as follows.

• Without PANC, IBO = 3.75 dB (η = 19%) with Ψ =4.5 dB for L = 1 active user, and IBO = 4.4 dB (η =16.2%) with Ψ = 4.4 dB for L = 2 active users.

• With PANC, IBO = 1.2 dB (η = 29.5%) with Ψ =2.55 dB for L = 1 active user, and IBO = 1.9 dB (η =26.4%) with Ψ = 2.58 dB for L = 2 users.

Note that for the PANC case, if IBO is fixed at 0 dB, Ψraises to 3.5 dB for L = 1 and to 3.2 dB for L = 2. Thissmall increment in Ψ gives an important improvement in thepower efficiency, which raises to 35%. We can then concludethat the optimal operation point for the PA can be moved toIBO = 0 dB, instead of the previous value (i.e., IBO = 1.2 dBfor L = 1 and IBO = 1.9 dB for L = 2). This is explained bythe specific shape of the total degradation curve which is almostflat when PANC is used in the low IBO region.

In order to verify the out-of-band distortion for the afore-mentioned operation points, we consider a WLAN system with512 carriers and an oversampling factor that is equal to 8.The PSD of the SSPA model is compared with the WLAN


Fig. 7. BER versus Eb/N0 of a P × L SDMA–OFDM system (P = 4) with 16-QAM for PANC with FD–TD channel estimation (T = 32 subcarriersper user), v(t) = 5 km/h, and using perfect CSI and LS detector, SSPA model with clipping ν = 4 dB. The curves are plotted for (a) L = 1 user,(b) L = 2 users, and (c) L = 3 users. Results obtained with linear PA and NL PA without PANC are included for reference.

transmit spectrum mask of a WLAN IEEE 802.11a system [33]in Fig. 9. The PSD of a linear PA is included as a reference. Theemployed clipping level is ν = 4 dB, and the back-off valuesare 0, 1.2, and 1.9 dB. As can be seen in the figure, the resultingout-of-band distortion from an SSPA that works in the optimalIBO points meets the transmit spectrum mask. Consequently,the PANC technique will give a reduced in-band distortion thatworks in the optimal work points with an adequate distortionover adjacent bands, as defined in the standard.

C. PANC in Time-Varying Channels

The system considered in this section uses 16-QAM on eachcarrier. We present the results obtained with PANC, assumingperfect CSI and using the FD–TD channel estimation approachproposed in Section V. The SSPA model uses a smoothingfactor p = 2 and ν = 4 dB. The FD–TD channel estimationemploys one iteration (i.e., one FD–TD cycle). An increase in

the number of iterations gives a small improvement in the esti-mation procedure, unnecessarily increasing the implementationcomplexity and delay. To compare the performance of the noveltechnique with other methods that combat NL PA effects, theLS-based PD in [30] was implemented. The PD was modeledwith a fifth-order polynomial.

Fig. 10 shows the MSE of the channel estimates versusSNR for different numbers of users. The alternatives evaluatedinclude the folowing.

1) FD–TD channel estimation for linear and NL PA models:These MSE curves give upper and lower bounds for thesystem performance.

2) FD–TD channel estimation using a seventh-order polyno-mial model of the PA model in the receiver: In this eval-uation, we show the robustness of the channel estimationmethodology to PA modeling errors.

3) FD–TD channel estimation with PANC: In this case, wecan evaluate the effect of MUI on the channel estimation


Fig. 8. Total degradation versus IBO of a P × L SDMA–OFDM system (P = 4) with 16-QAM for a receiver with and without PANC, v(t) = 5 km/h, LSdetector with FD–TD channel estimation (T = 32 subcarriers per user), v(t) = 5 km/h, SSPA model with clipping ν = 4 dB, and smoothness factor p = 2. Thecurves are plotted for P = 4 receive antennas, and L = 1 and L = 2 users. The power efficiency curve is obtained from [31].

Fig. 9. Out-of-band distortion of an OFDM system with 16-QAM, for dif-ferent values of IBO and SSPA model with p = 2 and ν = 4 dB. The resultsare plotted for optimal back-off values of IBO1 = 1.2 dB for L = 1 user andIBO2 = 1.9 dB for L = 2 users. The curves are seen to meet the requirementsthat are set by the WLAN transmit spectrum mask.

accuracy. This effect is clearly evident in these figures,given the lower MSE when the single-user case L = 1is considered. However, for a higher number of users,the BER performance is still suitable. These curves showresults that approach the values obtained when a linearPA is used.

4) FD–TD channel estimation with PD: This evaluationdemonstrates the performance of the PD without PANC.As can be observed, the results obtained with PANC(instead of PD) introduce improvements on the order of20 (for one user) to 10 dB (for four users).

5) FD–TD channel estimation with PD and PANC: Thisevaluation demonstrates that PD and PANC techniquesare complementary techniques and can be combined.

The curves in Fig. 10 show the advantage of using FD–TDchannel estimation with the PANC technique, even when com-bined with a PD.

The BER curves are found in Fig. 11. It can be observed thatan SNR gain that is larger than 2 dB is obtained for FD–TDchannel estimation with PANC when compared with a PD withFD–TD channel estimation, for L = 1. An even larger SNRgain is obtained for a higher number of users.

The performance degradation of the PD is related with thechannel estimation errors, and the results are even worse whenthe number of users is increased. On the other hand, in an NLsystem with PANC, the BER performance is slightly degradedwhen the number of users is increased.

Another issue that is shown in the BER results is that PANCis robust to PA modeling errors. The BER curves for PANCwith PA known and PANC with estimated PA parameters arealmost identical.

The results obtained by FD–TD channel estimation withPANC, and with the addition of PD, do not show a significantimprovement in the BER. However, this combination can beuseful because the PD provides a reduction in the out-of-banddistortion and PANC is useful for the in-band region.

VIII. CONCLUSION

A novel PANC technique in an SDMA–OFDM system is pre-sented, drastically reducing the harmful effects of PA nonlin-earities in the performance of the system. Significant levels ofpower efficiency improvements with reduced SNR degradationare obtained in SDMA–OFDM systems.

An upper bound for the BER for an SDMA–OFDM systemsubject to NL PA distortion was also derived. Simulation resultsagree well with the analytical derivation. The theoretical upperbound obtained can be used for a realistic system design,whereas the upper bound derived in [20], assuming a linearamplifier, may yield very optimistic BER values.


Fig. 10. MSE versus Eb/N0 of a P × L SDMA–OFDM system (P = 4) with 16-QAM for PANC and PD with FD–TD channel estimation (T = 16 subcarriersper user) and v(t) = 30 km/h, and LS detector, SSPA model with p = 2 and clipping ν = 4 dB. The curves are plotted for (a) L = 1 user, (b) L = 2 users,(c) L = 3, and (d) L = 4 users. Results obtained with perfectly known PA model and estimated PA parameters are included. The linear PA and NL PA withoutPANC are included for reference.

Finally, a channel estimation strategy that incorporates thechannel estimation into the PANC technique was proposed. Thechannel estimation algorithm operates in both the FD and TD. Itwas verified that, when incorporating the new channel estima-tion strategy into the PANC technique, the performance remainsclose to that of a perfectly known channel, thus preserving thesignificant improvement in BER levels that can be obtainedwhen compared with a solution that ignores the NL PA effects.The robustness of the technique to PA modeling errors was alsoverified.

To compare the performance of the novel technique withother methods that combat the PA nonlinearity effects, a PDwas also implemented. Simulation results show that PANCreaches better results than using a PD. The combination ofPANC and PD was also evaluated, and the results show thatthis combination can be useful, giving suitable BER levels anda reduced out-of-band distortion.

An interesting feature of PANC is that it implements thecomputationally intensive processing at the receiver. This is

an attractive scheme for uplink processing where the compu-tational burden is concentrated at the base station.

APPENDIX ALIMITER AND SSPA MODELS

The PA input signal can be represented in polar coordinatesas x = βejφ, and the output of the PA can be written as

g[x] =M [β] exp (j (φ+ P [β])) (32)

where M [β] represents the AM/AM conversion, and P [β] repre-sents the AM/PM conversion characteristics of the PA [34]. Theoutput of the PA can be written as

g[x] = KLx+ d (33)

where KL is a complex scalar that defines the linear gain, andd is an uncorrelated distortion term. Thus, we have

E[xd∗] = E [x (g[x]−KLx)∗] = 0 (34)


Fig. 11. BER versus Eb/N0 of a P × L SDMA–OFDM system (P = 4) with 16-QAM for PANC with FD–TD channel estimation (T = 16 subcarriersper user), v(t) = 30 km/h, and using perfect CSI and LS detector, SSPA model with clipping ν = 4 dB. The curves are plotted for (a) L = 1 user,(b) L = 2 users, and (c) L = 3 users. Results using perfectly known PA model and estimated PA model using a seventh-order polynomial, and linear PAand NL PA without PANC are included for reference.

from which we obtain the gain as

KL =E [xg∗[x]]E[xx∗]

. (35)

The NL distortion term can be calculated as

σ2d =

E[|g[x]|2

]− |KL|2

E [|x|2]

=

∫ +∞0 |g[u]|2p(u)du− |KL|2

σ2(36)

where g[u] =MS(β) is the PA transfer function, and p(u) isthe pdf of the OFDM signal, which, for large N , takes the formp(u) = (1/

√2πσ) exp(−(u2/2σ2)).

In case of the SSPA, the transfer characteristic is modeled by

MS(β) =β

[1 + (β/As)2p]1/2p

(37)

where the parameter p adjusts the smoothness of the transitionfrom the linear region to the saturation region. In the caseof the SSPA model, it is not possible to obtain a closed-form expression for the KL and σ2

d values. For values of plarger than 20, the SSPA model approaches the limiter modelgiven by

ML(β) ={β, β < As

As, β > As(38)

where As is the amplifier input saturation voltage.Using the limiter model, it is possible to obtain closed-form

solutions for the KL and σ2d values. From (35), and using the

PA model given by (38), we arrive at the following expressionfor KL [6]:

KL = 1− exp(−ν2) +12√πν erfc(ν) (39)


where ν is the clipping level. The power of the distortion noiseσ2d is obtained using (36), and it can be written as

σ2d =

∫ +∞0 |g[u]|2p(u)du−K2

L

σ2

=

∫ As

0 |u|2p(u)du+∫ +∞As

|As|2p(u)du−K2L

σ2

=σ2[1− exp(−ν2)−K2

L

]. (40)

APPENDIX BBER UPPER BOUND DERIVATION

Equation (16) can be rewritten as

Pe ≤1

(P − L)!γP−L+1

×∞∫

0

γP−L exp(−γ

γ

)exp

(−γ K2

L

σ2a + σ2

dγ

)dγ

=1

(P − L)!γP−L+1I (41)

where

I =

∞∫0

γk exp(−γ

γ

)f(γ)dγ,

k =P − L,

f(γ) = exp(−γ K2

L

σ2a + σ2

dγ

). (42)

To proceed, we use a Taylor expansion for f , i.e.,

f(γ) =∞∑

m=0

(−1)mm!

(γK2

L

σ2a + σ2

dγ

)m

. (43)

Now, integral I attains a sum form, where we need to evaluatethe integral

Im =

∞∫0

γke−γγ

(γK2

L

σ2a + σ2

dγ

)m

dγ. (44)

Using the substitution t = σ2dγ, we find that

Im =(1σ2d

)m+k+1

K2mL γ

∞∫0

tm+k

(σ2a + t)m

e−t/σ2dγ . (45)

We can use [21, eq. (8), p. 941] as follows:

∞∫0

xν−1(x+ β)−ρe−µxdx

= βν−ρ−1

2 µρ−ν−1

2 eβµ2 Γ(ν)W 1−ν−ρ

2 , ν−ρ2(βµ) (46)

where Γ(·) is the Gamma function, and W (·) is the Whittakerfunction defined as

Wλ,µ(z) =zµ+1/2e−z/2

Γ(µ− λ+ 1/2)

×∞∫

0

exp(−zt)tµ−λ−1/2(1 + t)λ+µ−1/2dt. (47)

By combining these equations, we can formulate the originalintegral as an expansion in Whittaker functions

Im =(1σ2d

)m+k+1

K2mL

(1σ2dγ

)k

× exp(

12σ2

dγ

)(m+ k)!W−k−2m

2 , k+12

(σ2a

σ2dγ

). (48)

Finally, substituting (48) in (41), the error probability can beexpressed as

Pe ≤1

(P − L)!γP−L+1

∞∑m=0

Im

=1

(P − L)!γP−L+1

(1σ2dγ

)P−Lexp

(1

2σ2dγ

)

×∞∑

m=0

(−1)m(m+ P − L)!m!

(1σ2d

)m+P−L+1

×K2mL W−(P−L)−2m

2 ,P−L+12

(σ2a

σ2dγ

). (49)

The Whittaker function can be calculated in MathematicaSymbolic Software using the Confluent Hypergeometric Func-tion, which is defined in (18). Equation (17) follows fromthe relation between the Whittaker function and the ConfluentHypergeometric functions. Hence

Wλ,µ(z) = exp−z/2 z1/2+µU1/2+µ−λ,1+2µ(z). (50)

APPENDIX CCAPACITY FOR LS RECEIVERS

The capacity is obtained by integrating (21) over the chi-square distribution as

CLS =

∞∫0

log2

(1 +

σ2jγ

σ2a + σ2

dγ/K2L

)

× 1(P − L)!γP−L+1

γP−L exp(−γ

γ

)dγ. (51)

Applying integration by parts and expressing the primi-tive function of the chi-square distribution as (ΓI(P − L+ 1,


γ/γ)/(P − L)!) [24, eq. (26.4.19)], where ΓI(·, ·) is theincomplete gamma function, the integral can be written as

CLS = log2

(1 +

σ2jγ

σ2a + σ2

d′γ,

)ΓI(P − L+ 1, γ/γ)

(P − L)!

∣∣∣∣∞

0

− 1ln(2)

∞∫0

ΓI(P − L+ 1, γ/γ)(P − L)!

g(γ)dγ

=T1 − T2 (52)

where σ2d′ = σ2

d/K2L, σ2

j = 1, and g(γ) = (σ2a/[σ

2a + (σ

2d′ +

1)γ][σ2a + σ2

d′γ]).The first term of (52) is equal to

T1 = log2

(1 +

1σ2d′

). (53)

In order to solve the second term T2, we assume that σ2d′ � 1

and γ � σa. The assumptions are for high SNR levels.Now, g(γ) can be accurately approximated as

g(γ) =σ2a

(σ2a + (σ2

d′ + 1) γ) (σ2a + σ2

d′γ)≈ σ2

a

γ (σ2a + σ2

d′γ).

(54)

Then, we express the incomplete gamma function ΓI(·, ·) usinga series expansion [35, Sec. 8.352, eq. (1)] as

ΓI(P−L+1, γ/γ)=(P − L)!

[1− exp

(γ

γ

) P−L∑k=0

(γ/γ)k

k!

].

(55)

Now, the term T2 can be written as

T2 =1

ln(2)

∞∫0

[1− exp

(γ

γ

)P−L∑k=0

(γ/γ)k

k!

]σ2a

γ (σ2a + σ2

d′γ)dγ

=

[1− exp

(γ

γ

)− exp

(γ

γ

)γ

γu(P − L)

− exp(γ

γ

) P−L∑k=2

(γ/γ)k

k!

]σ2a

γ (σ2a + σ2

d′γ))dγ

= I1 + I2 + I3 (56)

where u(P − L) is the unit step function defined as u(x) = 1for x > 0, and u(x) = 0 for x ≤ 0. After some manipulationand using [35], integral I1 is obtained as

I1 =1

ln(2)

∞∫0

[1− exp

(γ

γ

)]σ2a

γ (σ2a + σ2

d′γ)dγ

=1

ln(2)

[ln(

σ2a

σ2d′ γ

)− exp

(σ2a

σ2d′ γ

)Ei

(− σ2

a

σ2d′ γ

)+ γe

]

(57)

where γe is the Euler–Mascheroni constant (γe = 0.5772 . . .)[24], and Ei(·) is the exponential integral function [35].

This integral was solved using a series expansion for theexponential integral function [35, Sec. 8.214, eq. (4)] in orderto evaluate the integration limit in 0.

Integral I2 is obtained using [35, Sec. 3.352, eq. (4)]

I2 =1

ln(2)

∞∫0

exp(γ

γ

)1γ

σ2a

(σ2a + σ2

d′γ)u(P − L)dγ

=1

ln(2)σ2a

σ2d′ γ

exp(

σ2a

σ2d′ γ

)Ei

(− σ2

a

σ2d′ γ

)u(P − L).

(58)

Integral I3 is solved using [35, Sec. 3.384, eq. (10)]

I3 =1

ln(2)

∞∫0

exp(γ

γ

) P−L∑k=2

γk−1

γkk!σ2a

(σ2a + σ2

d′γ)dγ

=1

ln(2)exp

(σ2a

σ2d′ γ

) P−L∑k=2

σ2ka

γkkσ2kd′ΓI

(1− k,

σ2a

σ2d′ γ

).

(59)

Finally, the expression for the capacity can be expressed as

CLS ≈ log2

(1 +

1σ2d′

)− σ2

a

ln(2)

×[ln(β) + γe − exp (β)

×([1 + βu(P − L)]Ei(−β)

+P−L∑k=2

βkΓI(1− k, β)k

)](60)

where β = (σ2a/σ

2d′ γ).

ACKNOWLEDGMENT

The authors would like to thank Dr. R. Wichman andDr. J. Hämäläinen for their helpful suggestions.

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Fernando Gregorio (S’04) received the B.Sc. de-gree from the Universidad Tecnologica Nacional(UTN), Bahía Blanca, Argentina, and the M.Sc.degree in electrical engineering from the Univer-sidad Nacional del Sur (UNS), Bahía Blanca. Heis currently working toward the Ph.D. degree withthe Signal Processing Laboratory, Smart and NovelRadios (SMARAD) Centre of Excellence, HelsinkiUniversity of Technology, Espoo, Finland.

His research interests include power ampli-fier nonlinearities in MIMO–OFDM systems andmultiuser communications.

Stefan Werner (S’99–A’02–M’03) received theM.Sc. degree in electrical engineering from theRoyal Institute of Technology, Stockholm, Sweden,in 1998 and the D.Sc. degree (with honors) in elec-trical engineering from the Helsinki University ofTechnology (HUT), Espoo, Finland, in 2002.

He is currently a Senior Researcher with theSignal Processing Laboratory, Smart and Novel Ra-dios (SMARAD) Center of Excellence, HUT. Hisresearch interests are in multiuser communicationsand adaptive filtering.

Timo I. Laakso (SM’95) was born in Vantaa,Finland, on February 1, 1961. He received theM.Sc.(tech.), Lic.Sc.(tech.), and D.Sc.(tech.) degreesin electrical engineering from the Helsinki Univer-sity of Technology (HUT), Espoo, Finland, in 1987,1990, and 1991, respectively.

From 1992 to 1994, he was with the NokiaResearch Center, Helsinki, Finland, where he re-searched third-generation mobile communicationsystems. During 1994–1996, he was a Lecturer atthe University of Westminster, London, U.K. During

1996–2006, he was a Professor of signal processing in telecommunicationswith the Department of Electrical and Communications Engineering, HUT. Heis currently with the Signal Processing Laboratory, Smart and Novel Radios(SMARAD) Centre of Excellence, HUT. He is the author of approximately130 journal and conference publications. He is the holder of 16 patents.

Juan Cousseau (S’90–M’92–SM’00) received theB.Sc. degree in electrical engineering from the Uni-versidad Nacional del Sur (UNS), Bahía Blanca,Argentina, in 1983 and the M.Sc. and Ph.D. degreesin electrical engineering from the Universidade Fed-eral do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil,in 1989 and 1993, respectively.

Since 1984, he has been with the Departmentof Electrical and Computer Engineering and theDepartamento de Estudios de Postgrado y Edu-cación Contínua, UNS. He is currently a Senior

Researcher with the Consejo Nacional de Investigaciones Científicas y Técnicas(CONICET), Bahía Blanca, Argentina. He has been involved in scientific andindustry projects with research groups from Argentina, Brazil, Spain, and theUSA. He is the Coordinator of the Signal Processing and CommunicationLaboratory (LaPSyC), UNS. He was a Visiting Professor at the University ofCalifornia, Irvine, in 1999 and at the Signal Processing Laboratory, HelsinkiUniversity of Technology, Espoo, Finland, in 2004 and 2006. His researchinterests are in the areas of adaptive algorithms and statistical signal processingaddressed to high-speed digital communications.

Dr. Cousseau was the Chair of the IEEE Circuits and Systems Society(CASS), Argentine Chapter, from 1997 to 2000 and a member of the ExecutiveCommittee of the IEEE CASS during 2000/2001. He was part of the IEEESignal Processing Society Distinguished Lecturer Program in 2006.

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